---
tag: WHDZ3MK56B
kind: concept
name: Holonomy
msc: [53C29]
sources:
  - url: https://en.wikipedia.org/wiki/Holonomy
same_as:
  wikidata: Q907926
  wikipedia: Holonomy
---

## Summary

The holonomy of a [[connection|5ZYRRVE8BW]] records how [[parallel transport|TVAGN3D6W5]] around closed loops transforms the fibre at a base point.
These transformations form a group, the holonomy group, which is a Lie subgroup of the general linear group of the fibre.
The holonomy group measures curvature globally: it is trivial for flat connections on simply connected manifolds.
For Riemannian manifolds the possible holonomy groups are classified, and special holonomy groups correspond to special geometries such as Kähler, Calabi–Yau and $G_2$ manifolds.

## Explanations

### L1 human

Carry an arrow around a closed loop, keeping it parallel all the way.
When you come back, the arrow may have turned.
The holonomy is the set of all turns you can get from all loops at that point.
On a flat plane no loop turns the arrow, so the holonomy is trivial.
On a round sphere every rotation of the tangent plane can occur, so the holonomy is the rotation group $\mathrm{SO}(2)$.